cp's OEIS Frontend

A138378 Number of embedded coalitions in an n-person game.

%I A138378 #64 Dec 10 2024 09:58:18
%S A138378 1,3,10,37,151,674,3263,17007,94828,562595,3535027,23430840,163254885,
%T A138378 1192059223,9097183602,72384727657,599211936355,5150665398898,
%U A138378 45891416030315,423145657921379,4031845922290572,39645290116637023,401806863439720943,4192631462935194064
%N A138378 Number of embedded coalitions in an n-person game.
%C A138378 Same as A005493, apart from offset. - _R. J. Mathar_, Sep 23 2011
%C A138378 The strategic behavior of players depends crucially on the coalition structures of a game.
%D A138378 J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, New York, 1995.
%H A138378 Alois P. Heinz, <a href="/A138378/b138378.txt">Table of n, a(n) for n = 1..575</a>
%H A138378 E. T. Bell, <a href="http://www.jstor.org/stable/2300300">Exponential numbers</a>, Amer. Math. Monthly, 41 (1934), 411-419.
%H A138378 Gábor Czédli, <a href="https://arxiv.org/abs/2205.02294">Lattices with lots of congruence energy</a>, arXiv:2205.02294 [math.RA], 2022.
%H A138378 A. L. L. Gao, S. Kitaev, and P. B. Zhang. <a href="https://arxiv.org/abs/1605.05490">On pattern avoiding indecomposable permutations</a>, arXiv:1605.05490 [math.CO], 2016.
%H A138378 David W. K. Yeung, <a href="https://doi.org/10.1142/S0219198908001819">Recursive sequence identifying the number of embedded coalitions</a>, International Game Theory Review 10(2008), 129-136.
%F A138378 a(1) = combination(1,0) = 1, a(2) = combination(2,1) + combination(2,0)= 3, a(n) = {SUM(i=2 to n-1) combination(n,i)} * {SUM(j=1 to i-1) a(n)} + SUM(i=0 to n-1) combination(n,i), for n > 2.
%F A138378 G.f.: 1/U(0) where U(k)= 1 - x*(k+3) - x^2*(k+1)/U(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 11 2012
%F A138378 G.f.: 1/(U(0)-x) where U(k)= 1 - x - x*(k+1)/(1 - x/U(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Oct 12 2012
%F A138378 G.f.: -G(0)/x^2 where G(k) = 1 - 1/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Feb 08 2013
%F A138378 G.f.: Q(0)/x^2 -1/x^2, where Q(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*(k+1))*(1-x*(k+2))/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 10 2013
%F A138378 a(n) = Bell(n+1)-Bell(n) = Sum_{k=1..n} k*Stirling2(n,k). - _Alois P. Heinz_, May 11 2017
%F A138378 E.g.f.: (exp(x) - 1) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jan 26 2020
%e A138378 a(1) = combination(1,0) = 1,
%e A138378 a(2) = combination(2,1) + combination(2,0)= 3,
%e A138378 a(3) = combination(3,2)* a(1) + combination(3,2) + combination(3,1) + combination(3,0)= 10,
%e A138378 a(4) = combination(4,3)* {a(1) + a(2)} + combination(4,2)* a(1) + combination7(4,3)combination(4,2) + combination(4,1) + combination(4,0)= 37,
%e A138378 a(5) = combination(5,4)* {a(1) + a(2) + a(3)} combination(5,3)* {a(1) + a(2)} + combination(5,2)* a(1) + combination(5,4) + combination(5,3) + combination(5,2) + combination(5,1) + combination(5,0)= 151.
%p A138378 b:= proc(n, m) option remember;
%p A138378      `if`(n=0, m, m*b(n-1, m)+b(n-1, m+1))
%p A138378     end:
%p A138378 a:= n-> b(n, 0):
%p A138378 seq(a(n), n=1..24);  # _Alois P. Heinz_, Dec 10 2024
%t A138378 a[n_] := Sum[Binomial[n, j] BellB[j], {j, 0, n-1}];
%t A138378 Array[a, 24] (* _Jean-François Alcover_, Aug 19 2018 *)
%o A138378 (Magma) [&+[k*StirlingSecond(n, k): k in [1..n]]: n in [1..25]]; // _Vincenzo Librandi_, May 18 2019
%Y A138378 Cf. A000110, A005493, A048993, A138379.
%Y A138378 Column k=1 of A283424.
%K A138378 nonn
%O A138378 1,2
%A A138378 David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008